The World is Written in Beautiful Maths

Eight Suns

This is a timelapse multiple exposure photo of an arctic day, apparently titled “Six Suns” (even though there are 8 in the picture?) With credit to Circosatabolarc for posting the photo on Twitter, where I saw it. Photo by taken by Donald MacMillan of the Crocker Land Expedition, 1913-1917.

Attempting to resolve the name-suns discrepancy, I searched for “Six Suns” and found this photo, also taken by Donald MacMillan, from The Peary-MacMillian Arctic Museum, which actually shows six suns.

I hearby dub this photo “Eight Suns.”

A reverse image search turned up one more similar photo, a postcard titled “Midnight Sun and Moon,” taken at Fort McMurray on the Arctic Coast, sometime before 1943.

As you can see, above the arctic circle, the sun’s arc lies so low relative to the horizon that it appears to move horizontally across the sky. If you extended the photograph into a time-lapse movie, taken at the North Pole, you’d see the sun spiral upward from the Spring Equinox until it reaches 23.5 degrees above the horizon–about a quarter of the way to the top–on the Summer Solstice, and then spiral back down until the Fall Equinox, when it slips below the horizon for the rest of the year.

In other news, here’s a graph of size vs speed for three different classes of animals–flying, running, and swimming creatures–all of which show the same shape. “A general scaling law reveals why the largest animals are not the fastest” H/T NatureEcoEvo

I love this graph; it is a beautiful demonstration of the mathematics underlying bodily shape and design, not just for one class of animals, but for all of us. It is a rule that applies to all moving creatures, despite the fact that running, flying, and swimming are such different activities.

I assume similar scaling laws apply to mechanical and aggregate systems, as well.

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Homeschooling Corner: Summer Fun

Hello, everyone. I hope you have had a lovely summer. We ended up scaling back a bit on our regular schedule, doing about half as much formal “schoolwork” as usual and twice as much riding bikes and going to the playground.

Here are some of the books we found particularly useful/enjoyable this summer:

String, Straightedge, and Shadow: The Story of Geometry, by Julia E Diggins

This is my favorite book we read this summer.

I was looking for a book to introduce simple geometry and shape construction. Instead, I found this delightful history of geometry. It is appropriate for children who understand simple fractions, ratios, and the Pythagorean theorem, but it is not a mathematics textbook and only contains a few equations. (I’m still looking for an introduction to geometry, if anyone has any recommendations.)

This is a new edition of a book originally published in 1965, but its age isn’t really important because geometry hasn’t changed much in the intervening years.

The story begins with geometry in nature–the shapes of trees and flowers, spiderwebs and honeycombs–then develops a speculative account of how early stoneage humans might have become increasingly aware of and attuned to these shapes. Men saw the shapes of the sun and moon in the sky, and might have observed that an ox tied to a pole traced out a similar shape in the dirt.

Then Egyptian surveyors developed right triangles, used for measuring the corner of fields and pyramids. The Mesopotamians developed astronomy, and divided the circle into 360 degrees. Then came the Greeks–clever Thales, mystical Pythagoras, and practical Archimedes. And finally, at the end, Eratosthenes (who used geometry–literally, earth measuring–to measure the circumference of the Earth,) and a few paragraphs about Euclid.

Writing with Ease, by Susan Wise Bauer

There are many books and workbooks in this series, so you can pick the ones that best suit your child’s ability level. (The “look inside the book” feature is great for judging which level of textbook you want.)

I am sure these books are not everyone’s cup of tea. They may not be yours. But they were what we needed.

My eldest children are fairly different in writing needs, but I do not have time for separate curricula. One is a good speller, the other bad. One has acceptable handwriting, the other awful. One will write independently, the other hates writing and plays dead if I try to get them to write. These books have worked well for everyone. Spelling, handwriting, and general willingness to write have all improved.

Even if you aren’t homeschooling, this book might make a good supplement to your kids’ regular curriculum.

In science, we have been growing bacteria in petri dishes and looking at them under the microscope, with the help of Usborne Science and Experiments: The World of the Microscope (I think this is the same book, but cheaper.)

Petri dishes are cheap, agar is easy to make at home (it’s just like making jello,) and kids can learn things like “doorknobs are dirty” and “that’s why mom makes me wash my hands before dinner.”

Just be careful when handling large quantities of bacteria. Even if it’s normal household bacteria that you’re exposed to regularly, you’re not used to it in these quantities. The instructions recommend wearing gloves and safety goggles while handling bacteria and making slides out of them–and besides, kids like dressing up “like scientists” anyway.

The Super Source: Pattern Blocks and Geoboards

Our pattern blocks have been in the family for decades–passed down to me by my grandmother–but the geoboards are a new acquisition. I remember geoboards in elementary school–they sat behind the teacher’s desk and we never actually used them. I didn’t know what, exactly, geoboards were for, so I went ahead and got new workbooks for both them and the pattern blocks.

We are only a few lessons in, but so far I am very pleased with these. We have been talking about angles and measuring the degrees in different shapes with the pattern blocks–360 in a circle, 180 in a triangle, 720 in a hexagon, etc–which dovetails nicely with the geometry reading. The geoboards let us construct and examine a variety of different shapes, like right and equilateral triangles. The lesson plans are easy to use and the kids really enjoy them. Just watch out for rubber bands flying across the room.

Super Source makes workbooks for different grade levels, from K through 6th.

Learn to Program with Minecraft, by Craig Richardson

This book introduces Python, and is a nice step up from the Scratch workbooks. You may have to install a couple of programs, like Python and the API spigot, but the book walks you through this and it is not bad at all. There are then step-by-step instructions for making simple programs, along with bonus challenges to work out on your (or your kid’s) own.

The book covers strings, booleans, if statements, loops, etc, in kid-friendly ways. Best for people who already love Minecraft and can type.

Book Club: The Code Economy chs. 3-4: Machines and Computers

Machines are fascinating.

Imagine if we discovered, by some chance, in a previously unexplored niche of the world, a group of “people” who looked exactly like us, but had no technology at all: no fire, no pots, no textiles, not even an Acheulean hand axe. Let us further assume that they were not merely feral children who’d gotten lost in the woods, but an actual community that had sustained itself for generations, and that all attempts to introduce them to the art of tool-making failed. They could not, despite often watching others build fires or throw pots, make their own–much less understand and join in a modern economy. (A bulldog can learn to ride a skateboard, but it cannot learn to make a skateboard.)

What would we think of them? Would they be “human”? Even though they look like us, they could only live in houses and wear clothes if we gave those to them; if not given food, they would have to stay in their native environment and hunt.

It is hard to imagine a “human” without technology, nor explaining the course of human history without the advance of technology. The rise of trans-Atlantic discovery, trade, and migration that so fundamentally shaped the past 500 years cannot be explained without noting the development of superior European ships, maps, and navigational techniques necessary to make the journey. Why did Europe discover America and not the other way around? Many reasons, but fundamentally, because the Americans of 1492 didn’t have ships that could make the journey and the Europeans did.

The Romans were a surprisingly advanced society, technology wise, and even during the High Middle Ages, European technology continued to advance, as with the spread of wind and water mills. But I think it was the adoption of (Hindu-)Arabic numerals, popularized among mathematicians in the 1200s by Fibonacci, but only adopted more widely around the 1400s, that really allowed the Industrial and Information Revolutions to occur. (Roman numerals are just awful for any maths.)

An 18th century set of Napier’s Bones

From the Abacus and Fibonacci’s Liber Abaci, Napier developed his “bones,” the first European mechanical calculating machine, in 1617. These were followed by Pascal’s Calculator in 1642, the slide rule (early 1600s,) and Leibniz’s Stepped Reckoner in 1672. After that, progress on the adding machine problem was so rapid that it does not do to list all of the devices and prototypes devised, but we may mention Gaspard de Prony’s impressive logarithmic and trigonometric mathematical tables, for use by human “calculators”, and Babbage‘s analytical and difference machines. The Arithmometer, patented in 1820, was the first commercially successful mechanical calculator, used in many an office for nearly a century.

The history of mechanical computing devices wouldn’t be complete without reference to the parallel development of European clocks and watches, which pioneered the use of gears to translate movement into numbers, not to mention the development of an industry capable of manufacturing small, high-quality gears to reasonably high tolerances.

Given this context, I find our culture’s focus on Babbage–whose machines, while impressive, was never actually built–and his assistant Ada of Lovelace, a bit limited. Their contributions were interesting, but taken as a whole, the history is almost an avalanche of innovations.

Along the way, the computer has absorbed many technological innovations from outside computing–the Jacquard Loom pioneered the use of punch cards; the lightbulb pioneered the vacuum tubes that eventually filled Colossus and ENIAC.

But these early computers had a problem: vacuum tubes broke often.

During and immediately after World War II a phenomenon named “the tyranny of numbers” was noticed, that is, some computational devices reached a level of complexity at which the losses from failures and downtime exceeded the expected benefits.[2] Each Boeing B-29 (put into service in 1944) carried 300–1000 vacuum tubes and tens of thousands of passive components.[notes 4] The number of vacuum tubes reached thousands in advanced computers and more than 17,000 in the ENIAC (1946).[notes 5] Each additional component reduced the reliability of a device and lengthened the troubleshooting time.[2] Traditional electronics reached a deadlock and a further development of electronic devices required reducing the number of their components.

Also:

…the 1946 ENIAC, with over 17,000 tubes, had a tube failure (which took 15 minutes to locate) on average every two days. The quality of the tubes was a factor, and the diversion of skilled people during the Second World War lowered the general quality of tubes.[29]

The invention of the semiconductor further revolutionized computing–bringing us a long way from the abacus of yesterday.

Chapter 3 takes a break from the development of beautiful machines to examine their effects on humans. Auerswald writes:

By the twentieth century, the systematic approach to analyzing the division of labor that de Prony developed would have a name: management science. the first and foremost proponent of management science was Frederick Winslow Taylor, a child of privilege who found his calling in factories. …

This first experience of factory work gave Taylor an understanding of the habits of workers that was as intimate as it was, ultimately, unfavorable. Being highly organized and precise by nature, Taylor was appalled at the lax habits and absence of structure that characterized the early twentieth-century factory floor. … However, Taylor ultimately concluded that the blame did not lie with the workers but in the lack of rigorously considered management techniques. At the center of management, Taylor determined, was the capacity to precisely define the tasks of which a “job” was comprised.

What distinguished Taylor was his absolute conviction that worker could not be left on their own to define, much less refine, the tasks that comprised their work. He argued that authority must be fully vested in scientifically determined routine–that is to say, code.

Sounds hellish.

I know very little of management science beyond what can be found in Charlie Chaplin’s Modern Times. According to Wikipedia, Vladimir Lenin described Taylorism as a “‘scientific’ system of sweating” more work from laborers.[3] However, in Taylor’s defense, I don’t think the adoption of Taylorism ever resulted in the mass starvation of millions of people, so maybe Lenin should shut up was wrong. Further:

In the course of his empirical studies, Taylor examined various kinds of manual labor. For example, most bulk materials handling was manual at the time; material handling equipment as we know it today was mostly not developed yet. He looked at shoveling in the unloading of railroad cars full of ore; lifting and carrying in the moving of iron pigs at steel mills; the manual inspection of bearing balls; and others. He discovered many concepts that were not widely accepted at the time. For example, by observing workers, he decided that labor should include rest breaks so that the worker has time to recover from fatigue, either physical (as in shoveling or lifting) or mental (as in the ball inspection case). Workers were allowed to take more rests during work, and productivity increased as a result.[11]

Also:

By factoring processes into discrete, unambiguous units, scientific management laid the groundwork for automation and offshoring, prefiguring industrial process control and numerical control in the absence of any machines that could carry it out. Taylor and his followers did not foresee this at the time; in their world, it was humans that would execute the optimized processes. (For example, although in their era the instruction “open valve A whenever pressure gauge B reads over value X” would be carried out by a human, the fact that it had been reduced to an algorithmic component paved the way for a machine to be the agent.) However, one of the common threads between their world and ours is that the agents of execution need not be “smart” to execute their tasks. In the case of computers, they are not able (yet) to be “smart” (in that sense of the word); in the case of human workers under scientific management, they were often able but were not allowed. Once the time-and-motion men had completed their studies of a particular task, the workers had very little opportunity for further thinking, experimenting, or suggestion-making. They were forced to “play dumb” most of the time, which occasionally led to revolts.

While farming has its rhythms–the cows must be milked when the cows need to be milked, and not before or after; the crops must be harvested when they are ripe–much of the farmer’s day is left to his own discretion. Whether he wants to drive fence in the morning and hoe the peas in the afternoon, or attend to the peas first and the fence later is his own business. If he wants to take a nap or pause to fish during the heat of the day it is, again, his own business.

A factory can’t work like that. If the guy who is supposed to bolt the doors onto the cars can’t just wander off to eat a sandwich and use the restroom whenever he feels like it, nor can he decide that today he feels like installing windshields. Factories and offices allow many men to work together by limiting the scope of each one’s activities.

Is this algorithmisation of labor inhuman, and should we therefore welcome its mechanization and automation?

Why are there no Han Chinese Fields Medalists?

IQ by country

I am specifically referring to Han Chinese from the People’s Republic of China (hereafter simply called “China,”) but wanted to keep the title to a reasonable length.

There are about a billion Han Chinese. They make up about 90% of the PRC, and they have some of the highest average IQs on the planet, with particularly good math scores.

Of the 56 Fields Medals (essentially, the Nobel for Math) awarded since 1936, 12 (21%) have been French. 14 or 15 have been Jewish, or 25%-27%.

By contrast, 0 have been Han Chinese from China itself.

France is a country of 67.15 million people, of whom about 51 million are native French. The world has about 14-17.5 million Jews. China has about 1.37 billion people, of whom 91.51% are Han, or about 1.25 billion.

Two relatively Chinese people have won Fields medals:

Shing-Tung Yau was born in China, but is of Hakka ancestry (the Hakka are an Asian “market-dominant minority,”) not Han. His parents moved to Hong Kong when he was a baby; after graduating from the Chinese University of Hong Kong, he moved to the US, where he received his PhD from Berkley. Yau was a citizen of British-owned Hong Kong (not the People’s Republic of China), when he won the Fields Medal, in 1982; today he holds American citizenship.

Terence Tao, the 2006 recipient, is probably Han (Wikipedia does not list his ethnicity.) His father hailed from Shanghai, China, but moved to Hong Kong, where he graduated from medical school and met Tao’s mother, another Hong Kong-ian. Tao himself was born in Australia and later moved to the US. (Tao appears to be a dual Australian-American citizen.)

(With only 7.4 million people, Hong Kong is doing pretty well for itself in terms of Fields Medalists with some form of HK ancestry or citizenship.)

Since not many Fields Medals have been awarded, it is understandable why the citizens of small countries, even very bright ones, like Singapore, might not have any. It’s also understandable why top talent often migrates to places like Hong Kong, Australia, or the US. But China is a huge country with a massive pool of incredibly smart people–just look at Shanghai’s PISA scores. Surely Beijing has at least a dozen universities filled with math geniuses.

So where are they?

Is it a matter of funding? Has China chosen to funnel its best mathematicians into applied work? A matter of translation? Does the Fields Medal Committee have trouble reading papers written in Chinese? A matter of time? Did China’s citizens simply spent too much of the of the past century struggling at the edge of starvation to send a bunch of kids off to university to study math, and only recently achieved the level of mass prosperity necessary to start on the Fields path?

Whatever the causes of current under-representation, I have no doubt the next century will show an explosion in Han Chinese mathematical accomplishments.

Homeschooling Corner: A Mathematician’s Lament, by Paul Lockhart

Paul Lockhart’s A Mathematician’s Lament: How School Cheats us of our Most Fascinating and Imaginative Artform is a short but valuable book, easily finished in an afternoon.

Lockhart’s basic take is that most of us have math backwards. We approach (and thus teach) it as useful but not fun–something to be slogged through, memorized, and then avoided as much as possible. By contrast, Lockhart sees math as more fun than useful.

I do not mean that Lockhart denies the utility of balancing your checkbook or calculating how much power your electrical grid can handle, but most of the math actual mathematicians do isn’t practical. They do it because they enjoy it; they love making patterns with numbers and shapes. Just because paint has a very practical use in covering houses doesn’t mean we shouldn’t encourage kids to enjoy painting pictures; similarly, Lockhart wants kids to see mathematics as fun.

But wait, you say, what if this loosey-goosey, free-form, new math approach results in kids who spend a lot of time trying to re-derive pi from first principles but never really learning algebra? Lockhart would probably counter that most kids never truly master algebra anyway, so why make them hate it in the process? Should we only let kids who can paint like the Masters take art class?

If you and your kids already enjoy math, Lockhart may just reinforce what you already know, but if you’re struggling or math is a bore and a chore, Lockhart’s perspective may be just what you need to turn things around and make math fun.

For example: There are multiple ways to group the numbers during double-digit multiplication, all equally “correct”; the method you chose is generally influenced by things like your familiarity with double-digit multiplication and the difficulty of the problem. When I observed one of my kids making errors in multiplication because of incorrect regrouping, I showed them how to use a more expanded way of writing out the numbers to make the math clearer–promptly eliciting protests that I was “doing it wrong.” Inspired by Lockhart, I explained that “There is no one way to do math. Math is the art of figuring out answers, and there are many ways to get from here to there.” Learning how to use a particular approach—“Put the numbers here, here, and here and then add them”–is useful, but should not be elevated above using whatever approach best helps the child understand the numbers and calculate the correct answers.

The only difficulty with Lockhart’s approach is figuring out what to actually do when you sit down at the kitchen table with your kids, pencil and paper in hand. The book has a couple of sample lessons but isn’t a full k-12 curriculum. It’s easy to say, “I’m going to do a free-form curriculum that requires going to the library every day and uses every experience as a learning opportunity,” and rather harder to actually do it. With a set curriculum, you at least know, “Here’s what we’re going to do today.”

My own personal philosophy is that school time should be about 50% formal instruction and 50% open-ended exploration. Kids need someone to explain how the alphabet works and what these funny symbols on the math worksheet mean; they also need time to read fun books and play with numbers. They should memorize their times tables, but a good game can make times tables fun. In short, I think kids should have both a formal, straightforward curriculum or set of workbooks (I have not read enough math textbooks to recommend any particular ones,) and a set of math enrichment activities, like tangrams, pattern blocks, reading about Penrose the Mathematical Cat, or watching Numberphile on YouTube.

(Speaking of Penrose, I thought the chapter on binary went right over my kids’ heads, but yesterday they returned all of their answers in math class in binary, so I guess they picked up more than I gave them credit for.)

YouCubed.org is an interesting website I recently discovered. So far we’ve only done two of the activities, but they were cute and I suspect the website will make a useful addition to our lessons. If you’ve used it, I’d love to hear your thoughts on it.

That’s all for now. Happy learning!

Homeschooling Corner: Math ideas and manipulatives for younger grades

Archimedes

When you love a subject and your kids love it, too, it’s easy to teach. When you’re really not sure how to approach the subject or your kids hate it, it gets a lot trickier. (See: spelling.)

So I thought I’d make a list of some of our favorite math related materials–but please remember, all you really need for teaching math is a paper and pencil. (Or less–Archimedes did math with a stick and some sand!)

Countables

Little ones who are just learning to count and add benefit from having something concrete they can hold, touch, and move around when thinking about concepts like “two more” or “two less.”

You can count almost anything–pebbles, shells, acorns, pennies, Monopoly money, fingers–but having a box of dedicated, fun, colorful countables on hand is useful. My favorites:

Abacus. The abacus has the lovely advantage that all of its counters are on rods and so don’t get scattered around the room, stepped on and lost. I made my own abacus (inspired by commenter Dave‘s abacus) out of a shoe box, plastic beads, pipecleaners, and tape. You can count, add, subtract, multiply, divide, etc., on an abacus, but for your purposes you’ll just need to learn addition and subtraction.

Different abaci have different numbers and arrangements of beads. If your kids are still learning to count/mastering addition and subtraction up to ten (standard kindergarten goals,) I’d use an abacus with 9 beads per string. (Just like writing numbers, after you get to nine on the “ones” string, you raise up one bead on the “10” string.)

We adults tend to take place value for granted (“it’s obvious that we use the decimal system!”) but switching from column to column can be confusing for young kids. There’s no intuitive reason why 11 doesn’t = 2. The abacus helps increase awareness of place value (typically taught in first grade) because you simply run out of beads after 9 and have to switch to the next row.

Once kids have the basic idea, you can switch to a more advanced abacus like the Soroban. The top bead on the Soroban is worth 5, so students count 1-2-3-4, then click the 5 bead and clear the unit beads, then add unit beads to the five to count 6-7-8-9, then click one bead in the tens column and clear all of the beads in the unit and five column. My apologies if it sounds complicated; it really isn’t, it’s just a little tricky to put into words.

You can get abacus workbooks; I have not used any so I cannot review them but they look fun. Rather, I just use the abacus as a complement to the other math problems we are already doing. (I have read Mr. Green’s How to Use a Chinese Abacus, which was the only book my library had on the subject. It is a very good introduction aimed at adults.)

Counting Penguins

There is nothing magical about penguins; I just happen to like them. The set has 100 penguins in ten sets (distinguished by color) plus ten “ice bars” that hold ten penguins each. (Besides addition and subtraction,) I find these useful for introducing and visualizing multiplication , eg, 3 rows of 5 penguins = 3×5.

Counting Cubes

For bigger numbers, we have a bag of 1,000 interlocking cubes. Kids will want to just plain build with them, like Legos, which is fine–a fun treat after hard work. You can easily use these to represent 1s, 10s, and 100s (it takes a while to assemble a full 1,000 cube,) and to represent operations like 3x3x3, helping bridge both place value and multiplication. Legos work for this, too, though you’ll probably want to sort out ones that are all the same size and shape.

 

Pattern Blocks

(I think I’ve been incorrectly calling these tangrams, though the principles are similar.)

These pattern blocks are a family heirloom, sent to me by my grandmother upon the birth of my first child. I played with them when I was a child; my siblings played with them; now my children play with them. Someday I will pass them on to my grandchildren… but you can also get them on Amazon. (We use these with a book of pattern block activities that hails from the 80s; I am sure there are many good books of a similar nature published within the past couple of decades.

Apparently there are workbooks with pattern block activities aimed all the way up to 8th grade, but I have not read them and cannot comment on them.

Cuisenaire rods

We didn’t use cuisenaire rods when I was young, but I think I would have liked them. Similar to the tangrams pattern blocks, there are lots of interesting workbooks, games, and other activities you can do with these.

Building toys

Open-ended building toys (Legos, Tinker Toys, blocks, magnetic tiles) come in almost endless forms and can be used to build all sorts of geometric shapes.

Fractions

Fraction blocks and fraction circles are both handy.

Games

Almost any kids’ board game can be transformed into a math game by adding cards with math problems to be solved before completing a turn or using math dice. Your local games shop can help you find dice with numbers higher than six, or you can just tape paper onto an existing cube to make a custom die of your liking (like an + and – die). There are also tons of fun logic games; I pull these out whenever kids start getting restless.

Books

There are so many great math books, from Sir Cumference to Penrose, that I can’t hope to list them all. I encourage you to check out your library’s selection. Here are a few of my favorites:

The Adventures of Penrose the Mathematical Cat (plus sequels) makes a very pleasant enrichment portion of our daily maths. Each day we read one of Penrose’s stories (on subjects like Fibonacci numbers, primes, operations, etc) and do a short, related math activity.

Penrose is probably most appropriate for kids in mid to late elementary, not little ones just learning to count and add. (Note: the first story in the book was about binary, which flew over my kids’ heads.) Sir Cumference is more appropriate for younger learners.

Mathematicians are People, Too: biographies of great mathematicians. I’m not keen on the title, but my kids liked the chapter on Archimedes.

Balance Benders These workbooks come in different levels, from beginner to expert. Each puzzle presents students with a drawing of a balance with shapes on either side, and asks them to figure out, from a choice of answers, which statements about the shapes are true, eg “One circle equals two squares” after viewing a balance with two circles and four squares. (We also do logic puzzles and picture sudoku.)

 

Textbooks

I am not recommending any textbooks because I don’t have any idea which is the best. We don’t use a pre-packaged curriculum, because they tend to be expensive–instead I’ve just picked up a whole bunch of different math texts at the second hand shop and been gifted some lovely hand-me-downs from relatives. At this point I might have too many math books… I use 3 or 4 interchangably, depending on exactly which concepts we’re covering and whether I think the kids need more practice or not. I recently lucked into a volume of the “What your X Grader Needs to Know” series, and it gives a very nice overview of grade-level math expectations (among other things.)

Incidentally, the local public school math expectations appear to be:

Kindergarten: Reliably add and subtract the numbers 0-10; add small numbers to numbers between 10 and 20; be able to write all of the numbers from 0-20; count to 100.

1st grade: Place value; add and subtract one and two digit numbers with no regrouping.

2nd grade: Add and subtract multiple two an three-digit numbers.

I think they only explain regrouping in third grade.

In my experience, kids can do a lot more than that. These aren’t the standards I use in my classroom. But if you’re struggling to get your kindergartener to concentrate on their math worksheets, just remember: professional teachers don’t actually expect all that much at these ages. (And my kids don’t like doing a bunch of worksheet problems, either.)

Don’t sweat it. Do a few problems every day, if you can. Try teaching the same material from different angles, if necessary. Don’t be afraid to pull out pencil and paper and just make up a few problems and work through them together. Make patterns. Play games. Relax and have fun, because math at these ages really is beautiful.

Anthropology Friday: Numbers and the Making of Us, by Caleb Everett pt. 4

Yes, but which 25% of us is grape?

Welcome to our final post on Numbers and the Making of Us: Counting and the Course of Human Cultures, by Caleb Everett. Today I just want to highlight a few interesting passages.

On DNA:

For example, there is about 25% overlap between the human genome and that of grapes. (And we have fewer genes than grapes!) So some caution should be exercised before reading too much into percentages of genomic correspondence across species. I doubt, after all that you consider yourself one-quarter grape. … canine and bovine species generally exhibit about an 85% rate of genomic correspondence with humans. … small changes in genetic makeup can, among other influences, lead to large changes in brain size.

On the development of numbers:

Babylonian math homework

After all, for the vast majority of our species’ existence, we lived as hunters and gatherers in Africa … A reasonable interpretation of the contemporary distribution of cultural and number-system types, then, is that humans did not rely on complex number system for the bulk of their history. We can also reasonably conclude that transitions to larger, more sedentary, and more trade-based cultures helped pressure various groups to develop more involved numerical technologies. … Written numerals, and writing more generally, were developed first in the Fertile Crescent after the agricultural revolution began there. … These pressures ultimately resulted in numerals and other written symbols, such as the clay-token based numerals … The numerals then enabled new forms of agriculture and trade that required the exact discrimination and representation of quantities. The ancient Mesopotamian case is suggestive, then, of the motivation for the present-day correlation between subsistence and number types: larger agricultural and trade-based economies require numerical elaboration to function. …

Intriguingly, though, the same maybe true of Chinese writing, the earliest samples of which date to the Shang Dynasty and are 3,000 years old. The most ancient of these samples are oracle bones. These bones were inscribed with nuemerals quantifying such items as enemy prisoners, birds and animals hunted, and sacrificed animals. … Ancient writing around the world is numerically focused.

Changes in the Jungle as population growth makes competition for resources more intense and forces people out of their traditional livelihoods:

Consider the case of one of my good friends, a member of an indigenous group known as the Karitiana. … Paulo spent the majority of his childhood, in the 1980s and 1990s in the largest village of his people’s reservation. … While some Karitiana sought to make a living in nearby Porto Velho, many strived to maintain their traditional way of life on their reservation. At the time this was feasible, and their traditional subsistence strategies of hunting, gathering, and horticulture could be realistically practiced. Recently, however, maintaining their conventional way of life has become a less tenable proposition. … many Karitiana feel they have little choice but to seek employment in the local Brazilian economy… This is certainly true of Paulo. He has been enrolled in Brazilian schools for some time, has received some higher education, and is currently employed by a governmental organization. To do these things, of course, Paulo had to learn Portuguese grammar and writing. And he had to learn numbers and math, also. In short, the socioeconomic pressures he has felt to acquire the numbers of another culture are intense.

Everett cites a statistic that >90% of the world’s approximately 7,000 languages are endangered.

They are endangered primarily because people like Paulo are being conscripted into larger nation-states, gaining fluency in more economically viable languages. … From New Guinea to Australia to Amazonia and elsewhere, the mathematizing of people is happening.

On the advantages of different number systems:

Recent research also suggests that the complexity of some non-linguistic number systems have been under appreciated. Many counting boards and abaci that have been used, and are still in use across the world’s culture, present clear advantages to those using them … the abacus presents some cognitive advantages. That is because, research now suggests, children who are raised using the abacus develop a “mental abacus” with time. … According to recent cross-cultural findings, practitioners of abacus-based mathematical strategies outperform those unfamiliar with such strategies,a t least in some mathematical tasks. The use of the Soroban abacus has, not coincidentally, now been adopted in many schools throughout Asia.

The zero is a dot in the middle of the photo–earliest known zero, Cambodia

I suspect these higher math scores are more due to the mental abilities of the people using the abacus than the abacus itself. I have also just ordered an abacus.

… in 2015 the world’s oldest known unambiguous inscription of a circular zero was rediscovered in Cambodia. The zero in question, really a large dot, serves as a placeholder in the ancient Khmer numeral for 605. It is inscribed on a stone tablet, dating to 683 CE, that was found only kilometers from the faces of Bayon and other ruins of Angkor Wat and Angkor Thom. … the Maya also developed a written form for zero, and the Inca encoded the concept in their Quipu.

In 1202, Fibonacci wrote the Book of Calculation, which promoted the use of the superior Arabic (yes Hindu) numerals (zero included) over the old Roman ones. Just as the introduction of writing jump-started the Cherokee publishing industry, so the introduction of superior numerals probably helped jump-start the Renaissance.

Cities and the rise of organized religion:

…although creation myths, animistic practices, and other forms of spiritualism are universal or nearly universal, large-scale hierarchical religions are restricted to relatively few cultural lineages. Furthermore, these religions… developed only after people began living in larger groups and settlements because of their agricultural lifestyles. … A phalanx of scholars has recently suggested that the development of major hierarchical religions, like the development of hierarchical governments, resulted from the agglomeration of people in such places. …

Organized religious beliefs, with moral-enforcing deities and priest case, were a by-product of the need for large groups of people to cooperate via shared morals and altruism. As the populations of cultures grew after the advent of agricultural centers… individuals were forced to rely on shared trust with many more individuals, including non-kin, than was or is the case in smaller groups like bands or tribes. … Since natural selection is predicated on the protection of one’s genes, in-group altruism and sacrifice are easier to make sense of in bands and tribes. But why would humans in much larger populations–humans who have no discernible genetic relationship… cooperate with these other individuals in their own culture? … some social mechanism had to evolve so that larger cultures would not disintegrate due to competition among individuals and so that many people would not freeload off the work of others. One social mechanism that foster prosocial and cooperative behavior is an organized religion based on shared morals and omniscient deities capable of keeping track of the violation of such morals. …

When Moses descended from Mt. Sinai with his stone tablets, they were inscribed with ten divine moral imperatives. … Why ten? … Here is an eleventh commandment that could likely be uncontroversially adopted by many people: “thou shalt not torture.” … But then the list would appear to lose some of its rhetorical heft. “eleven commandments’ almost hints of a satirical deity.

Technically there are 613 commandments, but that’s not nearly as catchy as the Ten Commandments–inadvertently proving Everett’s point.

Overall, I found this book frustrating and repetitive, but there were some good parts. I’ve left out most of the discussion of the Piraha and similar cultures, and the rather fascinating case of Nicaraguan homesigners (“homesigners” are deaf people who were never taught a formal sign language but made up their own.) If you’d like to learn more about them, you might want to look up the book at your local library.

Anthropology Friday: Numbers and the Making of Us, by Caleb Everett, pt 3

Welcome back to our discussion of Numbers and the Making of Us: Counting and the Course of Human Cultures, by Caleb Everett.

The Pirahã are a small tribe (about 420) of Amazonian hunter-gatherers whose language is nearly unique: it has no numbers, and you can whistle it. Everett spent much of his childhood among the Piraha because his parents were missionaries, which probably makes him one of the world’s foremost non-Piraha experts on the Piraha.

Occasionally as a child I would wake up in the jungle to the cacophony of people sharing their dreams with one another–impromptu monologues followed by spurts of intense feedback. The people in question, a fascinating (to me anyhow) group known as the Piraha, are known to wake up and speak to their immediate neighbors at all hours of the night. … the voices suggested the people in the village were relaxed and completely unconcerned with my own preoccupations. …

The Piraha village my family lived in was reachable via a one-week sinuous trip along a series of Amazonian tributaries, or alternatively by a one-or flight in a Cessna single-engine airplane.

Piraha culture is, to say the least, very different from ours. Everett cites studies of Piraha counting ability in support of his idea that our ability to count past 3 is a culturally acquired process–that is, we can only count because we grew up in a numeric society where people taught us numbers, and the Piraha can’t count because they grew up in an anumeric society that not only lacks numbers, but lacks various other abstractions necessary for helping make sense of numbers. Our innate, genetic numerical abilities, (the ability to count to three and distinguish between small and large amounts,) he insists, are the same.

You see, the Piraha really can’t count. Line up 3 spools of thread and ask them to make an identical line, and they can do it. Line up 4 spools of thread, and they start getting the wrong number of spools. Line up 10 spools of thread, and it’s obvious that they’re just guessing and you’re wasting your time. Put five nuts in a can, then take two out and ask how many nuts are left: you get a response on the order of “some.”*

And this is not for lack of trying. The Piraha know other people have these things called “numbers.” They once asked Everett’s parents, the missionaries, to teach them numbers so they wouldn’t get cheated in trade deals. The missionaries tried for 8 months to teach them to count to ten and add small sums like 1 + 1. It didn’t work and the Piraha gave up.

Despite these difficulties, Everett insists that the Piraha are not dumb. After all, they survive in a very complex and demanding environment. He grew up with them; many of the are his personal friends and he regards them as mentally normal people with the exact same genetic abilities as everyone else who just lack the culturally-acquired skill of counting.

After all, on a standard IQ scale, someone who cannot even count to 4 would be severely if not profoundly retarded, institutionalized and cared for by others. The Piraha obviously live independently, hunt, raise, and gather their own food, navigate through the rainforest, raise their own children, build houses, etc. They aren’t building aqueducts, but they are surviving perfectly well outside of an institution.

Everett neglects the possibility that the Piraha are otherwise normal people who are innately bad at math.

Normally, yes, different mental abilities correlate because they depend highly on things like “how fast is your brain overall” or “were you neglected as a child?” But people also vary in their mental abilities. I have a friend who is above average in reading and writing abilities, but is almost completely unable to do math. This is despite being raised in a completely numerate culture, going to school, etc.

This is a really obvious and life-impairing problem in a society like ours, where you have to use math to function; my friend has been marked since childhood as “not cognitively normal.” It would be a completely invisible non-problem in a society like the Piraha, who use no math at all; in Piraha society, my friend would be “a totally normal guy” (or at least close.)

Everett states, explicitly, that not only are the Piraha only constrained by culture, but other people’s abilities are also directly determined by their cultures:

What is probably more remarkable about the relevant studies, though, is that they suggest that climbing any rungs of the arithmetic ladder requires numbers. How high we climb the ladder is not the result of our own inherent intelligence, but a result of the language we speak and of the culture we are born into. (page 136)

This is an absurd claim. Even my own children, raised in identically numerate environments and possessing, on the global scale, nearly identical genetics, vary in math abilities. You are probably not identical in abilities to your relatives, childhood classmates, next door neighbors, spouse, or office mates. We observe variations in mathematical abilities within cultures, families, cities, towns, schools, and virtually any group you chose that isn’t selected for math abilities. We can’t all do calculus just because we happen to live in a culture with calculus textbooks.

In fact, there is an extensive literature (which Everett ignores) on the genetics of intelligence:

Various studies have found the heritability of IQ to be between 0.7 and 0.8 in adults and 0.45 in childhood in the United States.[6][18][19] It may seem reasonable to expect that genetic influences on traits like IQ should become less important as one gains experiences with age. However, that the opposite occurs is well documented. Heritability measures in infancy are as low as 0.2, around 0.4 in middle childhood, and as high as 0.8 in adulthood.[7] One proposed explanation is that people with different genes tend to seek out different environments that reinforce the effects of those genes.[6] The brain undergoes morphological changes in development which suggests that age-related physical changes could also contribute to this effect.[20]

A 1994 article in Behavior Genetics based on a study of Swedish monozygotic and dizygotic twins found the heritability of the sample to be as high as 0.80 in general cognitive ability; however, it also varies by trait, with 0.60 for verbal tests, 0.50 for spatial and speed-of-processing tests, and 0.40 for memory tests. In contrast, studies of other populations estimate an average heritability of 0.50 for general cognitive ability.[18]

In 2006, The New York Times Magazine listed about three quarters as a figure held by the majority of studies.[21]

Thanks to Jayman

In plain speak, this means that intelligence in healthy adults is about 70-80% genetic and the rest seems to be random chance (like whether you were dropped on your head as a child or had enough iodine). So far, no one has proven that things like whole language vs. phonics instruction or two parents vs. one in the household have any effect on IQ, though they might effect how happy you are.

(Childhood IQ is much more amenable to environmental changes like “good teachers,” but these effects wear off as soon as children aren’t being forced to go to school every day.)

A full discussion of the scientific literature is beyond our current scope, but if you aren’t convinced about the heritability of IQ–including math abilities–I urge you to go explore the literature yourself–you might want to start with some of Jayman’s relevant FAQs on the subject.

Everett uses experiments done with the Piraha to support his claim that mathematical ability is culturally dependent, but this is dependent on is claim that the Piraha are cognitively identical to the rest of us in innate mathematical ability. Given that normal people are not cognitively identical in innate mathematical abilities, and mathematical abilities vary, on average, between groups (this is why people buy “Singapore Math” books and not “Congolese Math,”) there is no particular reason to assume Piraha and non-Piraha are cognitively identical. Further, there’s no reason to assume that any two groups are cognitively identical.

Mathematics only really got started when people invented agriculture, as they needed to keep track of things like “How many goats do I have?” or “Have the peasants paid their taxes?” A world in which mathematical ability is useful will select for mathematical ability; a world where it is useless cannot select for it.

Everett may still be correct that you wouldn’t be able to count if you hadn’t been taught how, but the Piraha can’t prove that one way or another. He would first have to show that Piraha who are raised in numerate cultures (say, by adoption,) are just as good at calculus as people from Singapore or Japan, but he cites no adoption studies nor anything else to this end. (And adoption studies don’t even show that for the groups we have studied, like whites, blacks, or Asians.)

Let me offer a cognitive contrast:

The Piraha are an anumeric, illiterate culture. They have encountered both letters and numbers, but not adopted them.

The Cherokee were once illiterate: they had no written language. Around 1809, an illiterate Cherokee man, Sequoyah, observed whites reading and writing letters. In a flash of insight, Sequoyah understand the concept of “use a symbol to encode a sound” even without being taught to read English. He developed his own alphabet (really a syllabary) for writing Cherokee sounds and began teaching it to others. Within 5 years of the syllabary’s completion, a majority of the Cherokee were literate; they soon had their own publishing industry producing Cherokee-language books and newspapers.

The Cherokee, though illiterate, possessed the innate ability to be literate, if only exposed to the cultural idea of letters. Once exposed, literacy spread rapidly–instantly, in human cultural evolution terms.

By contrast, the Piraha, despite their desire to adopt numbers, have not been able to do so.

(Yet. With enough effort, the Piraha probably can learn to count–after all, there are trained parrots who can count to 8. It would be strange if they permanently underperformed parrots. But it’s a difficult journey.)

That all said, I would like to make an anthropological defense of anumeracy: numeracy, as in ascribing exact values to specific items, is more productive in some contexts than others.

Do you keep track of the exact values of things you give your spouse, children, or close friends? If you invite a neighbor over for a meal, do you mark down what it cost to feed them and then expect them to feed you the same amount in return? Do you count the exact value of gifts and give the same value in return?

In Kabloona, de Poncin discusses the quasi-communist nature of the Eskimo economic system. For the Eskimo, hunter-gatherers living in the world’s harshest environment, the unit of exchange isn’t the item, but survival. A man whom you keep alive by giving him fish today is a man who can keep you alive by giving you fish tomorrow. Declaring that you will only give a starving man five fish because he previously gave you five fish will do you no good at all if he starves from not enough fish and can no longer give you some of his fish when he has an excess. The fish have, in this context, no innate, immutable value–they are as valuable as the life they preserve. To think otherwise would kill them.

It’s only when people have goods to trade, regularly, with strangers, that they begin thinking of objects as having defined values that hold steady over different transactions. A chicken is more valuable if I am starving than if I am not, but it has an identical value whether I am trading it for nuts or cows.

So it is not surprising that most agricultural societies have more complicated number systems than most hunter-gatherer societies. As Everett explains:

Led by Patience Epps of the University of Texas, a team of linguists recently documented the complexity of the number systems in many of the world’s languages. In particular, the researchers were concerned with the languages’ upper numerical limit–the highest quantity with a specific name. …

We are fond of coining new names for numbers in English, but the largest commonly used number name is googol (googolplex I define as an operation,) though there are bigger one’s like Graham’s.

The linguistic team in question found the upper numerical limits in 193 languages of hunter-gatherer cultures in Australia, Amazonia, Africa, and North America. Additionally, they examined the upper limits of 204 languages spoken by agriculturalists and pastoralists in these regions. They discovered that the languages of hunter-gatherer groups generally have low upper limits. This is particularly true in Australia and Amazonia, the regions with so-called pure hunter-gatherer subsistence strategies.

In the case of the Australian languages, the study in question observed that more than 80 percent are limited numerically, with the highest quantity represetned in such cases being only 3 or 4. Only one Australian language, Gamilaraay, was found to have an upper limit above 10, an dits highest number is for 20. … The association [between hunter-gathering and limited numbers] is also robust in South America and Amazonia more specifically. The languages of hunter-gatherer cultures in this region generally have upper limits below ten. Only one surveyed language … Huaorani, has numbers for quantities greater than 20. Approximately two-thirds of the languages of such groups in the region have upper limits of five or less, while one-third have an upper limit of 10. Similarly, about two-thirds of African hunter-gatherer languages have upper limits of 10 or less.

There are a few exceptions–agricultural societies with very few numbers, and hunter-gatherers with relatively large numbers of numbers, but:

…there are no large agricultural states without elaborate number systems, now or in recorded history.

So how did the first people develop numbers? Of course we don’t know, but Everett suggests that at some point we began associating collections of things, like shells, with the cluster of fingers found on our hands. One finger, one shell; five fingers, five shells–easy correspondences. Once we mastered five, we skipped forward to 10 and 20 rather quickly.

Everett proposes that some numeracy was a necessary prerequisite for agriculture, as agricultural people would need to keep track of things like seasons and equinoxes in order to know when to plant and harvest. I question this on the grounds that I myself don’t look at the calendar and say, “Oh look, it’s the equinox, I’d better plant my garden!” but instead look outside and say, “Oh, it’s getting warm and the grass is growing again, I’d better get busy.” The harvest is even more obvious: I harvest when the plants are ripe.

Of course, I live in a society with calendars, so I can’t claim that I don’t look at the calendar. I look at the calendar almost every day to make sure I have the date correct. So perhaps I am using my calendrical knowledge to plan my planting schedule without even realizing it because I am just so used to looking at the calendar.

“What man among you, if he has 100 sheep and has lost 1 of them, does not leave the 99 in the open pasture and go after the one which is lost until he finds it? When he has found it, he lays it on his shoulders, rejoicing.” Luke 15:3-5

Rather than develop numbers and then start planting barley and millet, I propose that humans first domesticated animals, like pigs and goats. At first people were content to have “a few,” “some,” or “many” animals, but soon they were inspired to keep better track of their flocks.

By the time we started planting millet and wheat (a couple thousand years later,) we were probably already pretty good at counting sheep.

Our fondness for tracking astronomical cycles, I suspect, began for less utilitarian reasons: they were there. The cycles of the sun, moon, and other planets were obvious and easy to track, and we wanted to figure out what they meant. We put a ton of work into tracking equinoxes and eclipses and the epicycles of Jupiter and Mars (before we figured out heliocentrism.) People ascribed all sorts of import to these cycles (“Communicator Mercury is retrograde in outspoken Sagittarius from December 3-22, mixing up messages and disrupting pre-holiday plans.”) that turned out to be completely wrong. Unless you’re a fisherman or sailor, the moon’s phases don’t make any difference in your life; the other planets’ cycles turned out to be completely useless unless you’re trying to send a space probe to visit them. Eclipses are interesting, but don’t have any real effects. For all of the effort we’ve put into astronomy, the most important results have been good calendars to keep track of dates and allow us to plan multiple years into the future.

Speaking of dates, let’s continue this discussion in a week–on the next Anthropology Friday.

*Footnote: Even though I don’t think the Piraha prove as much as Everett thinks they do, that doesn’t mean Everett is completely wrong. Maybe already having number words is (in the vast majority of cases) a necessary precondition for learning to count.

One potentially illuminating case Everett didn’t explore is how young children in numerate culture acquire numbers. Obviously they grow up in an environment with numbers, but below a certain age can’t really use them. Can children at these ages duplicate lines of objects or patterns? Or do they master that behavior only after learning to count?

Back in October I commented on Schiller and Peterson’s claim in Count on Math (a book of math curriculum ideas for toddlers and preschoolers) that young children must learn mathematical “foundation” concepts in a particular order, ie:

Developmental sequence is fundamental to children’s ability to build conceptual understanding. … The chapters in this book present math in a developmental sequence that provides children a natural transition from one concept to the next, preventing gaps in their understanding. …

When children are allowed to explore many objects, they begin to recognize similarities and differences of objects.

When children can determine similarities and differences, they can classify objects.

When children can classify objects, they can see similarities and difference well enough to recognize patterns.

When children can recognize, copy, extend and create patterns, they can arrange sets in a one-to-one relationship.

When children can match objects one to one, they can compare sets to determine which have more and which have less.

When children can compare sets, they can begin to look at the “manyness” of one set and develop number concepts.

This developmental sequence provides a conceptual framework that serves as a springboard to developing higher level math skills.

The Count on Math curriculum doesn’t even introduce the numbers 1-5 until week 39 for 4 year olds (3 year olds are never introduced to numbers) and numbers 6-10 aren’t introduced until week 37 for the 5 year olds!

Note that Schiller and Everett are arguing diametrical opposites–Everett says the ability to count to three and distinguish the “manyness” of sets is instinctual, present even in infants, but that the ability to copy patterns and match items one-to-one only comes after long acquaintance and practice with counting, specifically number words.

Schiller claims that children only develop the ability to distinguish manyness and count to three after learning to copy patterns and match items one-to-one.

As I said back in October, I think Count on Math’s claim is pure bollocks. If you miss the “comparing sets” day at preschool, you aren’t going to end up unable to multiply. The Piraha may not prove as much as Everett wants them to, but the neuroscience and animal studies he cites aren’t worthless. In general, I distrust anyone who claims that you must introduce this long a set of concepts in this strict an order just to develop a basic competency that the vast majority of people seem to acquire without difficulty.

Of course, Lynne Peterson is a real teacher with a real teacher’s certificate and a BA in … it doesn’t say, and Pam Schiller was Vice President of Professional Development for the Early childhood Division at McGraw Hill publishers and president of the Southern Early Childhood Association. She has a PhD in… it doesn’t say. Here’s some more on Dr. Schiller’s many awards. So maybe they know better than Everett, who’s just an anthropologist. But Everett has some actual evidence on his side.

But I’m a parent who has watched several children learn to count… and Schiller and Peterson are wrong.

Anthropology Friday: Numbers and the Making of Us, part 2

Welcome to part 2 of my review of Caleb Everett’s Numbers and the Making of Us: Counting and the Course of Human Cultures.

I was really excited about this book when I picked it up at the library. It has the word “numbers” on the cover and a subtitle that implies a story about human cultural and cognitive evolution.

Regrettably, what could have been a great books has turned out to be kind of annoying. There’s some fascinating information in here–for example, there’s a really interesting part on pages 249-252–but you have to get through pages 1-248 to get there. (Unfortunately, sometimes authors put their most interesting bits at the end so that people looking to make trouble have gotten bored and wandered off by then.)

I shall try to discuss/quote some of the book’s more interesting bits, and leave aside my differences with the author (who keeps reiterating his position that mathematical ability is entirely dependent on the culture you’re raised in.) Everett nonetheless has a fascinating perspective, having actually spent much of his childhood in a remote Amazonian village belonging to the Piraha, who have no real words for numbers. (His parents were missionaries.)

Which languages contain number words? Which don’t? Everett gives a broad survey:

“…we can reach a few broad conclusions about numbers in speech. First, they are common to nearly all of the world’s languages. … this discussion has shown that number words, across unrelated language, tend to exhibit striking parallels, since most languages employ a biologically based body-part model evident in their number bases.”

That is, many languages have words that translate essentially to “One, Two, Three, Four, Hand, … Two hands, (10)… Two Feet, (20),” etc., and reflect this in their higher counting systems, which can end up containing a mix of base five, 10, and 20. (The Romans, for example, used both base five and ten in their written system.)

“Third, the linguistic evidence suggests not only that this body-part model has motivated the innovation of numebers throughout the world, but also that this body-part basis of number words stretches back historically as far as the linguistic data can take us. It is evident in reconstruction of ancestral languages, including Proto-Sino-Tibetan, Proto-Niger-Congo, Proto-Autronesian, and Proto-Indo-European, the languages whose descendant tongues are best represented in the world today.”

Note, though, that linguistics does not actually give us a very long time horizon. Proto-Indo-European was spoken about 4-6,000 years ago. Proto-Sino-Tibetan is not as well studied yet as PIE, but also appears to be at most 6,000 years old. Proto-Niger-Congo is probably about 5-6,000 years old. Proto-Austronesian (which, despite its name, is not associated with Australia,) is about 5,000 years old.

These ranges are not a coincidence: languages change as they age, and once they have changed too much, they become impossible to classify into language families. Older languages, like Basque or Ainu, are often simply described as isolates, because we can’t link them to their relatives. Since humanity itself is 200,000-300,000 years old, comparative linguistics only opens a very short window into the past. Various groups–like the Amazonian tribes Everett studies–split off from other groups of humans thousands 0r hundreds of thousands of years before anyone started speaking Proto-Indo-European. Even agriculture, which began about 10,000-15,000 years ago, is older than these proto-languages (and agriculture seems to have prompted the real development of math.)

I also note these language families are the world’s biggest because they successfully conquered speakers of the world’s other languages. Spanish, Portuguese, and English are now widely spoken in the Americas instead of Cherokee, Mayan, and Nheengatu because Indo-European language speakers conquered the speakers of those languages.

The guy with the better numbers doesn’t always conquer the guy with the worse numbers–the Mongol conquest of China is an obvious counter. But in these cases, the superior number system sticks around, because no one wants to replace good numbers with bad ones.

In general, though, better tech–which requires numbers–tends to conquer worse tech.

Which means that even though our most successful language families all have number words that appear to be about 4-6,000 years old, we shouldn’t assume this was the norm for most people throughout most of history. Current human numeracy may be a very recent phenomenon.

“The invention of number is attainable by the human mind but is attained through our fingers. Linguistic data, both historical and current, suggest that numbers in disparate cultures have arisen independently, on an indeterminate range of occasions, through the realization that hands can be used to name quantities like 5 and 10. … Words, our ultimate implements for abstract symbolization, can thankfully be enlisted to denote quantities. But they are usually enlisted only after people establish a more concrete embodied correspondence between their finger sand quantities.”

Some more on numbers in different languages:

“Rare number bases have been observed, for instance, in the quaternary (base-4) systems of Lainana languages of California, or in the senary (base-6) systems that are found in southern New Guinea. …

Several languages in Melanesia and Polynesia have or once had number system that vary in accordance with the type of object being counted. In the case of Old High Fijian, for instance, the word for 100 was Bola when people were counting canoes, but Kora when they were counting coconuts. …

some languages in northwest Amazonia base their numbers on kinship relationships. This is true of Daw and Hup two related language in the region. Speakers of the former languages use fingers complemented with words when counting from 4 to 10. The fingers signify the quantity of items being counted, but words are used to denote whether the quantity is odd or even. If the quantity is even, speakers say it “has a brother,” if it is odd they state it “has no brother.”

What about languages with no or very few words for numbers?

In one recent survey of limited number system, it was found that more than a dozen languages lack bases altogether, and several do not have words for exact quantities beyond 2 and, in some cases, beyond 1. Of course, such cases represent a miniscule fraction of the world’s languages, the bulk of which have number bases reflecting the body-part model. Furthermore, most of the extreme cases in question are restricted geographically to Amazonia. …

All of the extremely restricted languages, I believe, are used by people who are hunter-gatherers or horticulturalists, eg, the Munduruku. Hunter gatherers typically don’t have a lot of goods to keep track of or trade, fields to measure or taxes to pay, and so don’t need to use a lot of numbers. (Note, however, that the Inuit/Eskimo have a perfectly normal base-20 counting system. Their particularly harsh environment appears to have inspired both technological and cultural adaptations.) But why are Amazonian languages even less numeric than those of other hunter-gatherers from similar environments, like central African?

Famously, most of the languages of Australia have somewhat limited number system, and some linguists previously claimed that most Australian language slack precise terms for quantities beyond 2…. [however] many languages on that continent actually have native means of describing various quantities in precise ways, and their number words for small quantities can sometimes be combined to represent larger quantities via the additive and even multiplicative usage of bases. …

Of the nearly 200 Australian languages considered in the survey, all have words to denote 1 and 2. In about three-quarters of the languages, however, the highest number is 3 or 4. Still, may of the languages use a word for “two” as a base for other numbers. Several of the languages use a word for “five” as a base, an eight of the languages top out at a word for “ten.”

Everett then digresses into what initially seems like a tangent about grammatical number, but luckily I enjoy comparative linguistics.

In an incredibly comprehensive survey of 1,066 languages, linguist Matthew Dryer recently found that 98 of them are like Karitiana and lack a grammatical means of marking nouns of being plural. So it is not particularly rare to find languages in which numbers do not show plurality. … about 90% of them, have a grammatical means through which speakers can convey whether they are talking about one or more than one thing.

Mandarin is a major language that has limited expression of plurals. According to Wikipedia:

The grammar of Standard Chinese shares many features with other varieties of Chinese. The language almost entirely lacks inflection, so that words typically have only one grammatical form. Categories such as number (singular or plural) and verb tense are frequently not expressed by any grammatical means, although there are several particles that serve to express verbal aspect, and to some extent mood.

Some languages, such as modern Arabic and Proto-Indo-European also have a “dual” category distinct from singular or plural; an extremely small set of languages have a trial category.

Many languages also change their verbs depending on how many nouns are involved; in English we say “He runs; they run;” languages like Latin or Spanish have far more extensive systems.

In sum: the vast majority of languages distinguish between 1 and more than one; a few distinguish between one, two, and many, and a very few distinguish between one, two, three, and many.

From the endnotes:

… some controversial claims of quadral markers, used in restricted contexts, have been made for the Austronesian languages Tangga, Marshallese, and Sursurunga. .. As Corbett notes in his comprehensive survey, the forms are probably best considered quadral markers. In fact, his impressive survey did not uncover any cases of quadral marking in the world’s languages.

Everett tends to bury his point; his intention in this chapter is to marshal support for the idea that humans have an “innate number sense” that allows them to pretty much instantly realize if they are looking at 1, 2, or 3 objects, but does not allow for instant recognition of larger numbers, like 4. He posits a second, much vaguer number sense that lets us distinguish between “big” and “small” amounts of things, eg, 10 looks smaller than 100, even if you can’t count.

He does cite actual neuroscience on this point–he’s not just making it up. Even newborn humans appear to be able to distinguish between 1, 2, and 3 of something, but not larger numbers. They also seem to distinguish between some and a bunch of something. Anumeric peoples, like the Piraha, also appear to only distinguish between 1, 2, and 3 items with good accuracy, though they can tell “a little” “some” and “a lot” apart. Everett also cites data from animal studies that find, similarly, that animals can distinguish 1, 2, and 3, as well as “a little” and “a lot”. (I had been hoping for a discussion of cephalopod intelligence, but unfortunately, no.)

How then, Everett asks, do we wed our specific number sense (1, 2, and 3) with our general number sense (“some” vs “a lot”) to produce ideas like 6, 7, and a googol? He proposes that we have no innate idea of 6, nor ability to count to 10. Rather, we can count because we were taught to (just as some highly trained parrots and chimps can.) It is only the presence of number words in our languages that allows us to count past 3–after all, anumeric people cannot.

But I feel like Everett is railroading us to a particular conclusion. For example, he sites neurology studies that found one part of the brain does math–the intraparietal suclus (IPS)–but only one part? Surely there’s more than one part of the brain involved in math.

About 5 seconds of Googling got me “Neural Basis of Mathematical Cognition,” which states that:

The IPS turns out to be part of the extensive network of brain areas that support human arithmetic (Figure 1). Like all networks it is distributed, and it is clear that numerical cognition engages perceptual, motor, spatial and mnemonic functions, but the hub areas are the parietal lobes …

(By contrast, I’ve spent over half an hour searching and failing to figure out how high octopuses can count.)

Moreover, I question the idea that the specific and general number senses are actually separate. Rather, I suspect there is only one sense, but it is essentially logarithmic. For example, hearing is logarithmic (or perhaps exponential,) which is why decibels are also logarithmic. Vision is also logarithmic:

The eye senses brightness approximately logarithmically over a moderate range (but more like a power law over a wider range), and stellar magnitude is measured on a logarithmic scale.[14] This magnitude scale was invented by the ancient Greek astronomer Hipparchus in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100.[14] Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.[15][16]

So many experiments have revealed logarithmic responses to stimuli that someone has formulated a mathematical “law” on the matter:

Fechner’s law states that the subjective sensation is proportional to the logarithm of the stimulus intensity. According to this law, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to logarithm of the actual intensity measured with an accurate nonhuman instrument.[3]

p = k ln ⁡ S S 0 {\displaystyle p=k\ln {\frac {S}{S_{0}}}\,\!}

The relationship between stimulus and perception is logarithmic. This logarithmic relationship means that if a stimulus varies as a geometric progression (i.e., multiplied by a fixed factor), the corresponding perception is altered in an arithmetic progression (i.e., in additive constant amounts). For example, if a stimulus is tripled in strength (i.e., 3 x 1), the corresponding perception may be two times as strong as its original value (i.e., 1 + 1). If the stimulus is again tripled in strength (i.e., 3 x 3 x 3), the corresponding perception will be three times as strong as its original value (i.e., 1 + 1 + 1). Hence, for multiplications in stimulus strength, the strength of perception only adds. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber’s law.[6][7]

In any logarithmic scale, small quantities–like 1, 2, and 3–are easy to distinguish, while medium quantities–like 101, 102, and 103–get lumped together as “approximately the same.”

Of course, this still doesn’t answer the question of how people develop the ability to count past 3, but this is getting long, so we’ll continue our discussion next week.

Review: Numbers and the Making of Us, by Caleb Everett

I’m about halfway through Caleb Everett’s Numbers and the Making of Us: Counting and the Course of Human Cultures. Everett begins the book with a lengthy clarification that he thinks everyone in the world has equal math abilities, some of us just happen to have been exposed to more number ideas than others. Once that’s out of the way, the book gets interesting.

When did humans invent numbers? It’s hard to say. We have notched sticks from the Paleolithic, but no way to tell if these notches were meant to signify numbers or were just decorated.

The slightly more recent Ishango, Lebombo, and Wolf bones (30,000 YA, Czech Republic) seem more likely to indicate that someone was at least counting–if not keeping track–of something.

The Ishango bone (estimated 20,000 years old, found in the Democratic Republic of the Congo near the headwaters of the Nile,) has three sets of notches–two sets total to 60, the third to 48. Interestingly, the notches are grouped, with both sets of sixty composed of primes: 19 + 17 + 13 + 11 and 9 + 19 + 21 + 11. The set of 48 contains groups of 3, 6, 4, 8, 10, 5, 5, and 7. Aside from the stray seven, the sequence tantalizingly suggests that someone was doubling numbers.

Ishango Bone

The Ishango bone also has a quartz point set into the end, which perhaps allowed it to be used for scraping, drawing, or etching–or perhaps it just looked nice atop someone’s decorated bone.

The Lebombo bone, (estimated 43-44,2000 years old, found near the border between South Africa and Swaziland,) is quite similar to the Ishango bone, but only contains 29 notches (as far as we can tell–it’s broken.)

I’ve seen a lot of people proclaiming “Scientists think it was used to keep track of menstrual cycles. Menstruating African women were the first mathematicians!” so I’m just going to let you in on a little secret: scientists have no idea what it was for. Maybe someone was just having fun putting notches on a bone. Maybe someone was trying to count all of their relatives. Maybe someone was counting days between new and full moons, or counting down to an important date.

Without a far richer archaeological assembly than one bone, we have no idea what this particular person might have wanted to count or keep track of. (Also, why would anyone want to keep track of menstrual cycles? You’ll know when they happen.)

The Wolf bone (30,000 years old, Czech Republic,) has received far less interest from folks interested in proclaiming that menstruating African women were the first mathematicians, but is a nice looking artifact with 60 notches–notches 30 and 31 are significantly longer than the others, as though marking a significant place in the counting (or perhaps just the middle of the pattern.)

Everett cites another, more satisfying tally stick: a 10,000 year old piece of antler found in the anoxic waters of Little Salt Spring, Florida. The antler contains two sets of marks: 28 (or possibly 29–the top is broken in a way that suggests another notch might have been a weak point contributing to the break) large, regular, evenly spaced notches running up the antler, and a much smaller set of notches set beside and just slightly beneath the first. It definitely looks like someone was ticking off quantities of something they wanted to keep track of.

Here’s an article with more information on Little Salt Spring and a good photograph of the antler.

I consider the bones “maybes” and the Little Salt Spring antler a definite for counting/keeping track of quantities.

Inca Quipu

Everett also mentions a much more recent and highly inventive tally system: the Incan quipu.

A quipu is made of knotted strings attached to one central string. A series of knots along the length of each string denotes numbers–one knot for 1, two for 2, etc. The knots are grouped in clusters, allowing place value–first cluster for the ones, second for the tens, third for hundreds, etc. (And a blank space for a zero.)

Thus a sequence of 2 knots, 4 knots, a space, and 5 knots = 5,402

The Incas, you see, had an empire to administer, no paper, but plenty of lovely alpaca wool. So being inventive people, they made do.

Everett then discusses the construction of names for numbers/base systems in different languages. Many languages use a combination of different bases, eg, “two twos” for four, (base 2,) “two hands” to signify 10 (base 5,) and from there, words for multiples of 10 or 20, (base 10 or 20,) can all appear in the same language. He argues convincingly that most languages derived their counting words from our original tally sticks: fingers and toes, found in quantities of 5, 10, and 20. So the number for 5 in a language might be “one hand”, the number for 10, “Two hands,” and the number for 20 “one person” (two hands + two feet.) We could express the number 200 in such a language by saying “two hands of one person”= 10 x 20.

(If you’re wondering how anyone could come up with a base 60 system, such as we inherited from the Babylonians for telling time, try using the knuckles of the four fingers on one hand [12] times the fingers of the other hand [5] to get 60.)

Which begs the question of what counts as a “number” word (numeral). Some languages, it is claimed, don’t have words for numbers higher than 3–but put out an array of 6 objects, and their speakers can construct numbers like “three twos.” Is this a number? What about the number in English that comes after twelve: four-teen, really just a longstanding mispronunciation of four and ten?

Perhaps a better question than “Do they have a word for it,” is “Do they have a common, easy to use word for it?” English contains the world nonillion, but you probably don’t use it very often (and according to the dictionary, a nonillion is much bigger in Britain than in the US, which makes it especially useless.) By contrast, you probably use quantities like a hundred or a thousand all the time, especially when thinking about household budgets.

Roman Numerals are really just an advanced tally system with two bases: 5 and 10. IIII are clearly regular tally marks. V (5) is similar to our practice of crossing through four tally marks. X (10) is two Vs set together. L (50) is a rotated V. C (100) is an abbreviation for the Roman word Centum, hundred. (I, V, X, and L are not abbreviations.) I’m not sure why 500 is D; maybe just because D follows C and it looks like a C with an extra line. M is short for Mille, or thousand. Roman numerals are also fairly unique in their use of subtraction in writing numbers, which few people do because it makes addition horrible. Eg, IV and VI are not the same number, nor do they equal 15 and 51. No, they equal 4 (v-1) and 6 (v+1,) respectively. Adding or multiplying large Roman numerals quickly becomes cumbersome; if you don’t believe me, try XLVII times XVIII with only a pencil and paper.

Now imagine you’re trying to run an empire this way.

You’re probably thinking, “At least those quipus had a zero and were reliably base ten,” about now.

Interestingly, the Mayans (and possibly the Olmecs) already had a proper symbol that they used for zero in their combination base-5/base-20 system with pretty functional place value at a time when the Greeks and Romans did not (the ancient Greeks were philosophically unsure about this concept of a “number that isn’t there.”)

(Note: given the level of sophistication of Native American civilizations like the Inca, Aztec, and Maya, and the fact that these developed in near total isolation, they must have been pretty smart. Their current populations appear to be under-performing relative to their ancestors.)

But let’s let Everett have a chance to speak:

Our increasingly refined means of survival and adaptation are the result of a cultural ratchet. This term, popularized by Duke University psychologist and primatologist Michael Tomasello, refers to the fact that humans cooperatively lock in knowledge from one generation to the next, like the clicking of a ratchet. In other word, our species’ success is due in large measure to individual members’ ability to learn from and emulate the advantageous behavior of their predecessors and contemporaries in their community. What makes humans special is not simply that we are so smart, it is that we do not have to continually come up with new solutions to the same old problems. …

Now this is imminently reasonable; I did not invent the calculus, nor could I have done so had it not already existed. Luckily for me, Newton and Leibniz already invented it and I live in a society that goes to great lengths to encode math in textbooks and teach it to students.

I call this “cultural knowledge” or “cultural memory,” and without it we’d still be monkeys with rocks.

The importance of gradually acquired knowledge stored in the community, culturally reified but not housed in the mind of any one individual, crystallizes when we consider cases in which entire cultures have nearly gone extinct because some of their stored knowledge dissipated due to the death of individuals who served as crucial nodes in their community’s knowledge network. In the case of the Polar Inuit of Northwest Greenland, population declined in the mid-nineteenth century after an epidemic killed several elders of the community. These elders were buried along with their tool sand weapons, in accordance with local tradition, and the Inuits’ ability to manufacture the tools and weapons in question was severely compromised. … As a result, their population did not recover until about 40 years later, when contact with another Inuit group allowed for the restoration of the communal knowledge base.

The first big advance, the one that separates us from the rest of the animal kingdom, was language itself. Yes, other animals can communicate–whales and birds sing; bees do their waggle dance–but only humans have full-fledged, generative language which allows us to both encode and decode new ideas with relative ease. Language lets different people in a tribe learn different things and then pool their ideas far more efficiently than mere imitation.

The next big leap was the development of visual symbols we could record–and read–on wood, clay, wax, bones, cloth, cave walls, etc. Everett suggests that the first of these symbols were likely tally marks such us those found on the Lebombo bone, though of course the ability to encode a buffalo on the wall of the Lascaux cave, France, was also significant. From these first symbols we developed both numbers and letters, which eventually evolved into books.

Books are incredible. Books are like external hard drives for your brain, letting you store, access, and transfer information to other people well beyond your own limits of memorization and well beyond a human lifetime. Books reach across the ages, allowing us to read what philosophers, poets, priests and sages were thinking about a thousand years ago.

Recently we invented an even more incredible information storage/transfer device: computers/the internet. To be fair, they aren’t as sturdy as clay tablets, (fired clay is practically immortal,) but they can handle immense quantities of data–and make it searchable, an incredibly important task.

But Everett tries to claim that cultural ratchet is all there is to human mathematical ability. If you live in a society with calculus textbooks, then you can learn calculus, and if you don’t, you can’t. Everett does not want to imply that Amazonian tribesmen with no words for numbers bigger than three are in any way less able to do math than the Mayans with their place value system and fancy zero.

But this seems unlikely for two reasons. First, we know very well that even in societies with calculus textbooks, not everyone can make use of them. Even among my own children, who have been raised with about as similar an environment as a human can make and have very similar genetics, there’s a striking difference in intellectual strengths and weaknesses. Humans are not identical in their abilities.

Moreover, we know that different mental tasks are performed in different, specialized parts of the brain. For example, we decode letters in the “visual word form area” of the brain; people whose VWAs have been damaged can still read, but they have to use different parts of their brains to work out the letters and they end up reading more slowly than they did before.

Memorably, before he died, the late Henry Harpending (of West Hunter) had a stroke while in Germany. He initially didn’t notice the stroke because it was located in the part of the brain that decodes letters into words, but since he was in Germany, he didn’t expect to read the words, anyway. It was only when he looked at something written in English later that day that he realized he couldn’t read it, and soon after I believe he passed out and was taken to the hospital.

Why should our brains have a VWA at all? It’s not like our primate ancestors did a whole lot of reading. It turns out that the VWA is repurposed from the part of our brain that recognizes faces :)

Likewise, there are specific regions of the brain that handle mathematical tasks. People who are better at math not only have more gray matter in these regions, but they also have stronger connections between them, letting the work together in harmony to solve different problems. We don’t do math by just throwing all of our mental power at a problem, but by routing it through specific regions of our brain.

Interestingly, humans and chimps differ in their ability to recognize faces and perceive emotions. (For anatomical reasons, chimps are more inclined to identify each other’s bottoms than each other’s faces.) We evolved the ability to recognize faces–the region of our brain we use to decode letters–when we began walking upright and interacting to each other face to face, though we do have some vestigial interest in butts and butt-like regions (“My eyes are up here.”) Our brains have evolved over the millenia to get better at specific tasks–in this case, face reading, a precursor to decoding symbolic language.

And there is a tremendous quantity of evidence that intelligence is at least partly genetic–estimates for the heritablity of intelligence range between 60 and 80%. The rest of the variation–the environmental part–looks to be essentially random chance, such as accidents, nutrition, or perhaps your third grade teacher.

So, yes, we absolutely can breed people for mathematical or linguistic ability, if that’s what the environment is selecting for. By contrast, if there have been no particular mathematical or linguistic section pressures in an environment (a culture with no written language, mathematical notation, and very few words for numbers clearly is not experiencing much pressure to use them), then you won’t select for such abilities. The question is not whether we can all be Newtons, (or Leibnizes,) but how many Newtons a society produces and how many people in that society have the potential to understand calculus, given the chance.

I do wonder why he made the graph so much bigger than the relevant part
Lifted gratefully from La Griffe Du Lion’s Smart Fraction II article

Just looking at the state of different societies around the world (including many indigenous groups that live within and have access to modern industrial or post-industrial technologies), there is clear variation in the average abilities of different groups to build and maintain complex societies. Japanese cities are technologically advanced, clean, and violence-free. Brazil, (which hasn’t even been nuked,) is full of incredibly violent, unsanitary, poorly-constructed favelas. Some of this variation is cultural, (Venezuela is doing particularly badly because communism doesn’t work,) or random chance, (Saudi Arabia has oil,) but some of it, by necessity, is genetic.

But if you find that a depressing thought, take heart: selective pressures can be changed. Start selecting for mathematical and verbal ability (and let everyone have a shot at developing those abilities) and you’ll get more mathematical and verbal abilities.

But this is getting long, so let’s continue our discussion next week.